×

zbMATH — the first resource for mathematics

Simple shear of a compressible quasilinear viscoelastic material. (English) Zbl 1423.74184
Summary: Fung’s theory of quasilinear viscoelasticity (QLV) was recently reappraised by the authors [“On nonlinear viscoelastic deformations: a reappraisal of Fung”s quasi-linear viscoelastic model”, Proc. R. Soc. A 479, No. 2166, Article ID 20140058 (2014; doi:10.1098/rspa.2014.0058)] in light of discussions in the literature of its apparent deficiencies. Due to the utility of the deformation of simple shear in a variety of applications, especially in experiment to deduce material properties, here QLV is employed to solve the problem of the simple shear of a nonlinear compressible quasilinear viscoelastic material. The effects of compressibility on the subsequent deformation and stress fields that result in this isochoric deformation are highlighted, and calculations of the dissipated energy associated with both a ‘ramp’ simple shear profile and oscillatory shear are given.

MSC:
74D10 Nonlinear constitutive equations for materials with memory
74D05 Linear constitutive equations for materials with memory
74B20 Nonlinear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abu Al-Rub, R.; Tehrani, A.; Darabi, M., Application of a large deformation nonlinear-viscoelastic viscoplastic viscodamage constitutive model to polymers and their composites, International Journal of Damage Mechanics, Available online, (2014)
[2] De Pascalis, R. (2010). The Semi-Inverse Method in solid mechanics: Theoretical underpinnings and novel applications, (Ph.D. thesis), Université Pierre et Marie Curie - Università del Salento.
[3] De Pascalis, R., Abrahams, I. D., & Parnell, W. J. (2014). On nonlinear viscoelastic deformations: A reappraisal of fung’s quasi-linear viscoelastic model, Proceedings of the Royal Society A, Vol. 479, p. 20140058.
[4] Drapaca, C.; Tenti, G.; Rohlf, K.; Sivaloganathan, S., A quasi-linear viscoelastic constitutive equation for the brain: application to hydrocephalus, Journal of Elasticity, 85, 65-83, (2006) · Zbl 1098.74040
[5] Fung, Y. C., Biomechanics: mechanical properties of living tissues, (1981), Springer Verlag New York
[6] Gilchrist, M.; Rashid, B.; Murphy, J.; Saccomandi, G., Quasi-static deformations of soft biological tissue, Mathematics and Mechanics of Solids, 18, 622-633, (2013)
[7] Johnson, G.; Livesay, G.; Woo, S.; Rajagopal, K., A single integral finite strain viscoelastic model of ligaments and tendons, Journal of Biomechanical Engineering, 118, 221-226, (1996)
[8] Levinson, M.; Burgess, I., A comparison of some simple constitutive relations for slightly compressible rubber-like materials, International Journal of Mechanical Science, 13, 563-572, (1971)
[9] Lion, A., A physically based method to represent the thermo-mechanical behaviour of elastomers, Acta Mechanica, 123, 1-25, (1997) · Zbl 0910.73019
[10] Peña, E.; Calvo, B.; Martínez, M.; Doblaré, M., An anisotropic visco-hyperelastic model for ligaments at finite strains. formulation and computational aspects, International Journal of Solids and Structures, 44, 760-778, (2007) · Zbl 1176.74043
[11] Pipkin, A.; Rogers, T., A non-linear integral representation for viscoelastic behaviour, Journal of the Mechanics and Physics of Solids, 16, 59-72, (1968) · Zbl 0158.43601
[12] Provenzano, P.; Lakes, R.; Corr, D.; Vanderby, R., Application of nonlinear viscoelastic models to describe ligament behavior, Biomechanics and Modeling in Mechanobiology, 1, 45-57, (2002)
[13] Rashid, B.; Destrade, M.; Gilchrist, M., Mechanical characterization of brain tissue in compression at dynamic strain rates, Journal of the Mechanical Behavior of Biomedical Materials, 10, 23-28, (2012)
[14] Simo, J., On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects, Computer Methods in Applied Mechanics and Engineering, 60, 153-173, (1987) · Zbl 0588.73082
[15] Waldron, W.; Wineman, A., Shear and normal stress effects in finite circular shear of a compressible nonlinear viscoelastic solid, International Journal of Non-linear Mechanics, 31, 345-369, (1996) · Zbl 0863.73023
[16] Wineman, A., Nonlinear viscoelastic solids—a review, Mathematics and Mechanics of Solids, 14, 300-366, (2009) · Zbl 1197.74021
[17] Wineman, A.; Waldron, W., Yield-like response of a compressible nonlinear viscoelastic solid, Journal of Rheology, 39, 401-423, (1995)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.